Integrand size = 41, antiderivative size = 383 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {2 \left (161 a^2 b B+63 b^3 B+15 a^3 C+5 a b^2 (49 A+29 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (56 a^3 b B-56 a b^3 B-10 a^2 b^2 (7 A-C)+15 a^4 C-5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 a^3 A \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (35 A b^2+56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 (7 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \] Output:
2/105*(161*B*a^2*b+63*B*b^3+15*a^3*C+5*a*b^2*(49*A+29*C))*(a+b*cos(d*x+c)) ^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/b/d/((a+b*cos (d*x+c))/(a+b))^(1/2)-2/105*(56*B*a^3*b-56*B*a*b^3-10*a^2*b^2*(7*A-C)+15*a ^4*C-5*b^4*(7*A+5*C))*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d *x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/b/d/(a+b*cos(d*x+c))^(1/2)+2*a^3*A*((a+b *cos(d*x+c))/(a+b))^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(b/(a+b) )^(1/2))/d/(a+b*cos(d*x+c))^(1/2)+2/105*(35*A*b^2+56*B*a*b+15*C*a^2+25*C*b ^2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/d+2/35*(7*B*b+5*C*a)*(a+b*cos(d*x+c) )^(3/2)*sin(d*x+c)/d+2/7*C*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/d
Result contains complex when optimal does not.
Time = 12.86 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.37 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {\frac {4 \left (105 a^3 B+119 a b^2 B+45 a^2 b (7 A+3 C)+5 b^3 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (161 a^2 b B+63 b^3 B+15 a^3 (14 A+C)+5 a b^2 (49 A+29 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 i \left (161 a^2 b B+63 b^3 B+15 a^3 C+5 a b^2 (49 A+29 C)\right ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {-\frac {b (1+\cos (c+d x))}{a-b}} \csc (c+d x) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right )}{a b^2 \sqrt {-\frac {1}{a+b}}}+2 \sqrt {a+b \cos (c+d x)} \left (70 A b^2+154 a b B+90 a^2 C+65 b^2 C+6 b (7 b B+15 a C) \cos (c+d x)+15 b^2 C \cos (2 (c+d x))\right ) \sin (c+d x)}{210 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2)*Sec[c + d*x],x]
Output:
((4*(105*a^3*B + 119*a*b^2*B + 45*a^2*b*(7*A + 3*C) + 5*b^3*(7*A + 5*C))*S qrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/S qrt[a + b*Cos[c + d*x]] + (2*(161*a^2*b*B + 63*b^3*B + 15*a^3*(14*A + C) + 5*a*b^2*(49*A + 29*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, ( c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 5*a*b^2*(49*A + 29*C))*Sqrt[-((b*(-1 + Cos[c + d* x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x]*(-2*a* (a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[ Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b^2* Sqrt[-(a + b)^(-1)]) + 2*Sqrt[a + b*Cos[c + d*x]]*(70*A*b^2 + 154*a*b*B + 90*a^2*C + 65*b^2*C + 6*b*(7*b*B + 15*a*C)*Cos[c + d*x] + 15*b^2*C*Cos[2*( c + d*x)])*Sin[c + d*x])/(210*d)
Time = 3.30 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.04, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.585, Rules used = {3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left ((7 b B+5 a C) \cos ^2(c+d x)+(7 A b+5 C b+7 a B) \cos (c+d x)+7 a A\right ) \sec (c+d x)dx+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int (a+b \cos (c+d x))^{3/2} \left ((7 b B+5 a C) \cos ^2(c+d x)+(7 A b+5 C b+7 a B) \cos (c+d x)+7 a A\right ) \sec (c+d x)dx+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((7 b B+5 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(7 A b+5 C b+7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+7 a A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (35 A a^2+\left (15 C a^2+56 b B a+35 A b^2+25 b^2 C\right ) \cos ^2(c+d x)+\left (35 B a^2+70 A b a+40 b C a+21 b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {a+b \cos (c+d x)} \left (35 A a^2+\left (15 C a^2+56 b B a+35 A b^2+25 b^2 C\right ) \cos ^2(c+d x)+\left (35 B a^2+70 A b a+40 b C a+21 b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (35 A a^2+\left (15 C a^2+56 b B a+35 A b^2+25 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (35 B a^2+70 A b a+40 b C a+21 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {\left (105 A a^3+\left (15 C a^3+161 b B a^2+5 b^2 (49 A+29 C) a+63 b^3 B\right ) \cos ^2(c+d x)+\left (105 B a^3+45 b (7 A+3 C) a^2+119 b^2 B a+5 b^3 (7 A+5 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (105 A a^3+\left (15 C a^3+161 b B a^2+5 b^2 (49 A+29 C) a+63 b^3 B\right ) \cos ^2(c+d x)+\left (105 B a^3+45 b (7 A+3 C) a^2+119 b^2 B a+5 b^3 (7 A+5 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {105 A a^3+\left (15 C a^3+161 b B a^2+5 b^2 (49 A+29 C) a+63 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (105 B a^3+45 b (7 A+3 C) a^2+119 b^2 B a+5 b^3 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\int -\frac {\left (105 a^3 A b-\left (15 C a^4+56 b B a^3-10 b^2 (7 A-C) a^2-56 b^3 B a-5 b^4 (7 A+5 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}+\frac {\int \frac {\left (105 a^3 A b-\left (15 C a^4+56 b B a^3-10 b^2 (7 A-C) a^2-56 b^3 B a-5 b^4 (7 A+5 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {\int \frac {105 A b a^3+\left (-15 C a^4-56 b B a^3+10 b^2 (7 A-C) a^2+56 b^3 B a+5 b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {105 A b a^3+\left (-15 C a^4-56 b B a^3+10 b^2 (7 A-C) a^2+56 b^3 B a+5 b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {105 A b a^3+\left (-15 C a^4-56 b B a^3+10 b^2 (7 A-C) a^2+56 b^3 B a+5 b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\int \frac {105 A b a^3+\left (-15 C a^4-56 b B a^3+10 b^2 (7 A-C) a^2+56 b^3 B a+5 b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {105 a^3 A b \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\left (15 a^4 C+56 a^3 b B-10 a^2 b^2 (7 A-C)-56 a b^3 B-5 b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}+\frac {2 \left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {105 a^3 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (15 a^4 C+56 a^3 b B-10 a^2 b^2 (7 A-C)-56 a b^3 B-5 b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {105 a^3 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\left (15 a^4 C+56 a^3 b B-10 a^2 b^2 (7 A-C)-56 a b^3 B-5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {105 a^3 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\left (15 a^4 C+56 a^3 b B-10 a^2 b^2 (7 A-C)-56 a b^3 B-5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {105 a^3 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (15 a^4 C+56 a^3 b B-10 a^2 b^2 (7 A-C)-56 a b^3 B-5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\frac {105 a^3 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 \left (15 a^4 C+56 a^3 b B-10 a^2 b^2 (7 A-C)-56 a b^3 B-5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\frac {105 a^3 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 \left (15 a^4 C+56 a^3 b B-10 a^2 b^2 (7 A-C)-56 a b^3 B-5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {2 \left (15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\frac {210 a^3 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (15 a^4 C+56 a^3 b B-10 a^2 b^2 (7 A-C)-56 a b^3 B-5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}\right )\right )+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec [c + d*x],x]
Output:
(2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + ((2*(7*b*B + 5*a*C)* (a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (((2*(161*a^2*b*B + 63*b^ 3*B + 15*a^3*C + 5*a*b^2*(49*A + 29*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE [(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ( (-2*(56*a^3*b*B - 56*a*b^3*B - 10*a^2*b^2*(7*A - C) + 15*a^4*C - 5*b^4*(7* A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/ (a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (210*a^3*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Co s[c + d*x]]))/b)/3 + (2*(35*A*b^2 + 56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d))/5)/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1712\) vs. \(2(369)=738\).
Time = 56.59 (sec) , antiderivative size = 1713, normalized size of antiderivative = 4.47
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1713\) |
| parts | \(\text {Expression too large to display}\) | \(2019\) |
Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x,me thod=_RETURNVERBOSE)
Output:
-2/105*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*C* b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+(-168*B*b^4-480*C*a*b^3-360*C* b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(140*A*b^4+392*B*a*b^3+168*B* b^4+360*C*a^2*b^2+480*C*a*b^3+280*C*b^4)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+ 1/2*c)+(-70*A*a*b^3-70*A*b^4-154*B*a^2*b^2-196*B*a*b^3-42*B*b^4-90*C*a^3*b -180*C*a^2*b^2-170*C*a*b^3-80*C*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2* c)+70*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b )/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2+35 *A*b^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b) /(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+245*A*(sin( 1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2 )*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2-245*A*(sin(1/2* d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*El lipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^3-105*A*a^3*(sin(1/2*d* x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Elli pticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b-56*B*(sin(1/2*d*x+1/2*c) ^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(co s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b+56*B*a*(sin(1/2*d*x+1/2*c)^2)^( 1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2 *d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3+161*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*...
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c ),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x +c),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c ),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*sec(d*x + c), x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c ),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*sec(d*x + c), x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\cos \left (c+d\,x\right )} \,d x \] Input:
int(((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/c os(c + d*x),x)
Output:
int(((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/c os(c + d*x), x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) b^{2} c +2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )d x \right ) a b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )d x \right ) b^{3}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )d x \right ) a^{2} c +3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )d x \right ) a \,b^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )d x \right ) a^{3} \] Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x)
Output:
3*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x),x)*a**2*b + int(s qrt(cos(c + d*x)*b + a)*cos(c + d*x)**4*sec(c + d*x),x)*b**2*c + 2*int(sqr t(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x),x)*a*b*c + int(sqrt(cos (c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x),x)*b**3 + int(sqrt(cos(c + d *x)*b + a)*cos(c + d*x)**2*sec(c + d*x),x)*a**2*c + 3*int(sqrt(cos(c + d*x )*b + a)*cos(c + d*x)**2*sec(c + d*x),x)*a*b**2 + int(sqrt(cos(c + d*x)*b + a)*sec(c + d*x),x)*a**3